Philosophy 145
Professor Amy Kind
Spring 2008
Homework for April 8
· Read pp. 76-85
· Do the Exercise on Quantifier Rules (handout)
· Prove S87-S93. I strongly recommend that you use the Logic Daemon to do so. (http://logic.tamu.edu)
· Do #23-46; 61; 63-66 in exercise 3.2 in the book, as indicated below
Notes for Exercise 3.2
Do the following sentences in Exercise 3.2: 23-46; 61; 63-66. Note that some of these sentences may be ambiguous, in which case you should give both readings of the sentence. Also, for this exercise, be on the lookout for the hidden predicate:
Pa: a is a person
Translation Schemes
#23–29
s: Shamu
Ta: a is a trick
Wa: a is a whale
Cab: a can do b
Note: read #29 as “If some whale can do a trick, any whale can do a trick.”
#30-42; 61; 63-66
g: Godzilla
b: Bambi
Aab: a ate b
#43-46
Sabg: a said b to g
Exercise – The Conditions on our Quantifier Rules
Some of the following proofs involve illegal applications of our quantifier rules. For each proof, indicate whether any of the moves is illegal, and for any illegal move, indicate why.
i. Fa |- $xFx
1 (1) Fa A
1 (2) $xFx 1 $I
ii. Fa |- "xFx
1 (1) Fa A
1 (2) "xFx 1 "I
iii. "xFx |- Fa
1 (1) "xFx A
1 (1) Fa 1 "E
iv. $xFx |- Fa
1 (1) $xFx A
2 (2) Fa A [for $E on 1]
1 (3) Fa 1,2 $E (2)
v. $xFx |- "xFx
1 (1) $xFx A
2 (2) Fa A [for $E on 1]
2 (3) "xFx 2 "I
1 (4) "xFx 1,3 $E (2)
vi. "xFx |- $xFx
1 (1) "xFx A
1 (2) Fa 1 "E
1 (3) $xFx 2 $I
vii. "xFx ® "xGx |- "x(Fx ® Gx)
1 (1) "xFx ® "xGx A
2 (2) Fa A [for ®I]
2 (3) "xFx 2 "I
1,2 (4) "xGx 1,3 ®E
1,2 (5) Ga 4 "E
1 (6) Fa ® Ga 5 ®I (2)
1 (7) "x(Fx ® Gx) 6 "I
viii. "x(Fx ® Gx) |- "xFx ® "xGx
1 (1) "x(Fx ® Gx) A
2 (2) "xFx A [for ®I]
2 (3) Fa 2 "E
1 (4) Fa ® Ga 1 "E
1,2 (5) Ga 3,4 ®E
1,2 (6) "xGx 5 "I
1 (7) "xFx ® "xGx 6 ®I (2)
ix. $x(Fx & Gx) |- $xFx & $xGx
1 (1) $x(Fx & Gx) A
2 (2) Fa & Ga A [for $E on 1]
2 (3) Fa 2 &E
2 (4) $xFx 3 $I
1 (5) $xFx 1,4 $E (2)
2 (6) Ga 2 &E
2 (7) $xGx 6 $I
1 (8) $xGx 1,7 $E (2)
1 (9) $xFx & $xGx 5,8 &I
x. $xFx & $xGx |- $x(Fx & Gx)
1 (1) $xFx & $xGx A
1 (2) $xFx 1 &E
3 (3) Fa A [for $E on 2]
1 (4) $xGx 1 &E
5 (5) Ga A [for $E on 4]
3,5 (6) Fa & Ga 3,5 &I
3,5 (7) $x(Fx & Gx) 6 $I
1,3 (8) $x(Fx & Gx) 4,7 $E (5)
1 (9) $x(Fx & Gx) 2,8 $E (3)